Locally Nilpotent Groups and Hyperfinite Equivalence Relations
Locally Nilpotent Groups and Hyperfinite Equivalence Relations
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局部幂零群和超有限等价关系
DOI：
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发表时间：
2013
期刊：
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通讯作者：
Brandon Seward
中科院分区：
文献类型：
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作者：
S. Schneider;Brandon Seward
超有限等价关系理论中一个长期存在的开放性问题是：可数可服从群的Borel作用所产生的轨道等价关系是否超有限。本文证明了当作用群局部幂零时，这个问题有一个正解。这推广了之前Gao-Jackson关于阿贝尔群和Jackson-Kechris-Louveau关于有限生成的幂零乘有限群的结果。我们的证明是基于局部幂零群的粗糙几何性质的混合以及对Gao-Jackson机制的适应。
A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we show that this question has a positive answer when the acting group is locally nilpotent. This extends previous results obtained by Gao-Jackson for abelian groups and by Jackson-Kechris-Louveau for finitely generated nilpotent-by-finite groups. Our proof is based on a mixture of coarse geometric properties of locally nilpotent groups together with an adaptation of the Gao-Jackson machinery.